UC Berkeley

The Langlands-Kottwitz method seeks to understand Shimura varieties in terms of automorphic forms by deriving a trace formula for the cohomology of Shimura varieties which can be compared to the automorphic trace formula. This method was pioneered by Langlands [Lan77, Lan79], and developed further by Kottwitz in [Kot90, Kot92] in the case of Shimura varieties of PEL type with good reduction.

Igusa varieties were introduced in their modern form by Harris-Taylor [HT01] in the course of studying the bad reduction of certain simple Shimura varieties. The relation between Igusa varieties and Shimura varieties was expanded to PEL type by Mantovan [Man04, Man05], and their study of the cohomology of Igusa varieties was expanded to PEL type and streamlined in the Langlands-Kottwitz style by Shin [Shi09, Shi10].

Following the generalization of Mantovan's work to Hodge type by Hamacher and Hamacher-Kim [Ham19, HK19], we carry out the Langlands-Kottwitz method for Igusa varieties of Hodge type, generalizing the work of [Shi09]. That is, we derive a trace formula for the cohomology of Igusa varieties suitable for eventual comparison with the automorphic trace formula.

In order to carry out this method, we also formulate and prove an analogue of the Langlands-Rapoport conjecture for Igusa varieties of Hodge type, building off work of Kisin [Kis17] for Shimura varieties of Hodge and abelian type. Our subsequent arguments on the way to our trace formula are inspired by the techniques of Kisin-Shin-Zhu [KSZ21] using the Langlands-Rapoport conjecture to develop a trace formula for Shimura varieties of Hodge and abelian type.